Browsing by Author "Shahmorad, Sedaghat"

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Existence, Uniqueness and Blow-Up of Solutions for Generalized Auto-Convolution Volterra Integral Equations
(Elsevier Science inc, 2024) Mostafazadeh, Mahdi; Shahmorad, Sedaghat; Erdogan, Fevzi
In this paper, our intention is to investigate the blow-up theory for generalized auto -convolution Volterra integral equations (AVIEs). To accomplish this, we will consider certain conditions on the main equation. This will establish a framework for our analysis, ensuring that the solution of the equation exists uniquely and is positive. Firstly, we analyze the existence and uniqueness of a local solution for a more general class of AVIEs (including the proposed equation in this paper) under certain hypotheses. Subsequently, we demonstrate the conditions under which this local solution blows up at a finite time. In other words, the solution becomes unbounded at that time. Furthermore, we establish that this blow-up solution can be extended to an arbitrary interval on the non -negative real line, thus referred to as a global solution. These results are also discussed for a special case of generalized AVIEs in which the kernel functions are taken as positive constants.
Mean Square Stability of Numerical Method for Stochastic Volterra Integral Equations with Double Weakly Singular Kernels
(ISCI-Inst Scientific Computing & Information, 2025) Rouz, Omid farkhondeh; Shahmorad, Sedaghat; Erdogan, Fevzi
The main goal of this paper is to develop an improved stochastic 9-scheme as a numerical method for stochastic Volterra integral equations (SVIEs) with double weakly singular kernels and demonstrate that the stability of the proposed scheme is affected by the kernel parameters. To overcome the low computational efficiency of the stochastic 9-scheme, we employed the sum-of-exponentials (SOE) approximation. Then, the mean square stability of the proposed scheme with respect to a convolution test equation is studied. Additionally, based on the stability conditions and the explicit structure of the stability matrices, analytical and numerical stability regions are plotted and compared with the split-step 9-method and the 9-Milstein method. The results confirm that our approach aligns significantly with the expected physical interpretations.
Review of Recursive and Operational Approaches of the Tau Method With a New Extension
(Springer Heidelberg, 2023) Shahmorad, Sedaghat; Talaei, Younes; Tunc, Cemil
This is a review paper that briefly represents the recursive and operational approaches to the Tau method on solving ordinary differential and integro-differential equations with suitable initial or boundary conditions, and we discuss a new extension of the method on solving a class of Abel Volterra integral equations which can be also used for solving fractional differential equations. Extension of height and canonical polynomials are introduced. Illustrative examples are given in each case to clarify the performance and structural properties of the method.
Solving a Class of Auto-Convolution Volterra Integral Equations Via Differential Transform Method
(Univ Guilan, 2025) Tari, Abolfazl; Shahmorad, Sedaghat; Mostafazadeh, Mahdi; Erdogan, Fevzi
The aim of this paper is to solve a class of auto-convolution Volterra integral equations by the well-known differential transform method. The analytic property of solution and convergence of the method under some assumptions are discussed and some illustrative examples are given to clarify the theoretical results, accuracy and performance of the proposed method.